Can I make a deep image with OpenEXR?
Blender is not the right tool to do this. You can do almost everything in Blender, but sometimes coding and doing it yourself is required.

You cannot obtain the deep image directly, because there is no renderer (as my knowledge goes) that can output that (a volumetric output).

You get multi-channel output at most (beauty pass, z-depth, shadow pass,..) - and this is what most 3d apps will let you save:

Save all render passes to a single EXR image

Some will let you save this as deep image, for example MentalRay.

To get your image with deep pixels, you would have to create it from all the rendered images. To be able to do this inside Blender, you would have to use the OpenEXR source code and code it yourself and run it from Blender through python bindings. Some compositing applications offer creation of openEXR deep images, like Nuke for example.

A deep image is nothing else than volumetric data structure in camera space. You cannot 'render' such data in Blender, but you can create them (smoke, fire, ..) and export as Alembic (in global space). From there it might be easier to handle such data externally.

If you need to convert particles or geometry into volumetric data structures, Houdini is a good tool to do that.